| 释义 |
- Properties
- General linear groups
- References
In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive group G over a global field and a finite-dimensional complex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by {{harvs|txt|authorlink=Robert Langlands|last=Langlands|year1=1967|year2=1970|year3=1971}}. {{harvtxt|Borel|1979}} and {{harvtxt|Arthur|Gelbart|1991}} gave surveys of automorphic L-functions.PropertiesAutomorphic L-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases). The L-function L(s,π,r) should be a product over the places v of F of local L functions. L(s,π,r) = Π L(s,πv,rv) Here the automorphic representation π=⊗πv is a tensor product of the representations πv of local groups. The L-function is expected to have an analytic continuation as a meromorphic function of all complex s, and satisfy a functional equation L(s,π,r) = ε(s,π,r)L(1 – s,π,r∨) where the factor ε(s,π,r) is a product of "local constants" ε(s,π,r) = Π ε(s,πv,rv, ψv) almost all of which are 1. General linear groups{{harvtxt|Godement|Jacquet|1972}} constructed the automorphic L-functions for general linear groups with r the standard representation (so-called standard L-functions) and verified analytic continuation and the functional equation, by using a generalization of the method in Tate's thesis. Ubiquitous in the Langlands Program are Rankin-Selberg products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the Langlands–Shahidi method.In general, the Langlands functoriality conjectures imply that automorphic L-functions of a connected reductive group are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions. References- {{Citation | last1=Arthur | first1=James | last2=Gelbart | first2=Stephen | author2-link = Stephen Gelbart | editor1-last=Coates | editor1-first=John | editor2-last=Taylor | editor2-first=M. J. | title=L-functions and arithmetic (Durham, 1989) | url=http://www.claymath.org/cw/arthur/pdf/automorphic-L.pdf | publisher=Cambridge University Press | series=London Math. Soc. Lecture Note Ser. | isbn=978-0-521-38619-7 | doi=10.1017/CBO9780511526053.003 | mr=1110389 | year=1991 | volume=153 | chapter=Lectures on automorphic L-functions | pages=1–59}}
- {{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | editor1-last=Borel | editor1-first=Armand | editor1-link=Armand Borel | editor2-last=Casselman | editor2-first=W. | title=Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 | url=http://www.ams.org/publications/online-books/pspum332-index | publisher=American Mathematical Society | location=Providence, R.I. | series=Proc. Sympos. Pure Math. | isbn=978-0-8218-1437-6 | mr=546608 | year=1979 | volume= XXXIII | chapter=Automorphic L-functions | pages=27–61 | doi=10.1090/pspum/033.2/546608}}
- {{Citation | last1=Cogdell | first1=James W. | last2=Kim | first2=Henry H. | last3=Murty | first3=Maruti Ram | title=Lectures on automorphic L-functions | url=https://books.google.com/books?id=jb3ZCp0-MQsC | publisher=American Mathematical Society | location=Providence, R.I. | series=Fields Institute Monographs | isbn=978-0-8218-3516-6 | mr=2071722 | year=2004 | volume=20}}
- {{Citation | last1=Gelbart | first1=Stephen | author1-link = Stephen Gelbart | last2=Piatetski-Shapiro | first2=Ilya | last3=Rallis | first3=Stephen | title=Explicit constructions of automorphic L-functions | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-17848-4 | doi=10.1007/BFb0078125 | mr=892097 | year=1987 | volume=1254}}
- {{Citation | last1=Godement | first1=Roger | author1-link=Roger Godement | last2=Jacquet | first2=Hervé | title=Zeta functions of simple algebras | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-05797-0 | doi=10.1007/BFb0070263 | mr=0342495 | year=1972 | volume=260}}
- {{Citation | last1=Jacquet | first1=H. | last2=Piatetski-Shapiro | first2=I. I. | last3=Shalika | first3=J. A. | title=Rankin-Selberg Convolutions | year=1983 | journal=Amer. J. Math. | volume=105 | pages=367–464 | doi=10.2307/2374264}}
- {{citation|last=Langlands|first=Robert|title=Letter to Prof. Weil|year=1967|url=http://publications.ias.edu/rpl/section/21}}
- {{Citation | last1=Langlands | first1=R. P. | title=Lectures in modern analysis and applications, III | url=http://publications.ias.edu/rpl/section/21 | publisher=Springer-Verlag | location=Berlin, New York | series= Lecture Notes in Math | isbn=978-3-540-05284-5 | doi=10.1007/BFb0079065 | mr=0302614 | year=1970 | volume=170 | chapter=Problems in the theory of automorphic forms | pages=18–61}}
- {{Citation | last1=Langlands | first1=Robert P. | title=Euler products | origyear=1967 | url=http://publications.ias.edu/rpl/paper/37 | publisher=Yale University Press | isbn=978-0-300-01395-5 | mr=0419366 | year=1971}}
- {{Citation | last1=Shahidi | first1=F. | title=On certain "L"-functions | year=1981 | journal=Amer. J. Math. | volume=103 | pages=297–355 | doi=10.2307/2374219}}
3 : Automorphic forms|Zeta and L-functions|Langlands program |